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Degeneracy means the same energy. For example, with hydrogen, every subshell in the same energy level will be degenerate, because there is no electron-electron repulsion countering the pull of the nucleus, as there is only one electron. However, with multi-electron atoms, the subshells within the same energy level become degenerate, having different energy levels in the order of s<p<d<f. Subshells of lower energy are able to "shield" or offset the pull of the nucleus for subshells of higher energy (within the same shell). The reason these subshells have differing energies depends on how well they are able to penetrate the nucleus. The s subshell has no nodes, meaning it has a non-zero probability of being found in the nucleus. Therefore, it is able to penetrate the nucleus unlike the other subshells, therefore more effectively shielding the other electrons. However, with each increasing subshell type, there is more and more angular momentum driving the electron away if it gets to the nucleus, so they are less and less able to get close to the nucleus, staying at further at a higher energy level instead.
As far as degeneracy is concerned it is important to know that electrons in the same hydrogen atom have the same energy level. However, multi electron systems are not degenerate as with different sub shells there are different energies.
For hydrogen, all orbitals share the same energy, so the degeneracy of an orbital simply depends on how many exist. For multi-electron atoms, only orbitals that share the same l quantum number have the same energy, so the degeneracy of an orbital would then depend on how many possible orbitals is possible for the indicated l quantum number.
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